lecture_12

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Lecture 12
The Effective Action as a Generating
Functional
Here we will rederive the effective potential by making use of the fact that the effective
action is a generating functional of the one-particle-irreducible (1PI) diagrams. To see
this, we start from the definition of the generating functional of all Feynman diagrams as
Z[J ] = eiW [J ] , (12.1)
where, as we showed earlier, W [J ] is the generating functional of the connected Feynman
diagrams. In other words, since the n-point function is generated from Z[J ] as
G(n)(x1, . . . , xn) =
1
Z[0]
(−i)n δ
n
δJ(x1) . . . δJ(xn)
Z[J ]
∣∣∣∣
J=0
, (12.2)
we can write them as
Z[J ] = Z[0]
∞∑
n=0
in
n!
∫
dx1 . . . dxnG
(n)(x1, . . . , xn) J(x1) . . . J(xn) . (12.3)
Likewise, the connected generating functional iW [J ] can be expanded in terms of the
connected n-point correlation functions G
(n)
c (x1, . . . , xn) as
iW [J ] =
∞∑
n=0
in
n!
∫
dx1 . . . dxnG
(n)
c (x1, . . . , xn) J(x1) . . . J(xn) . (12.4)
On the other hand, we recall the definition of the effective action as
1
2 LECTURE 12. THE EFFECTIVE ACTION AS A GENERATING FUNCTIONAL
Γ[φc] = W [J ]−
∫
d4x J(x)φc(x) , (12.5)
where φc(x) = 〈0|φ(x)|0〉. As seen in the previous lecture, we have that
δΓ[φc]
δφc(x)
= −J(x) , δW [J ]
δJ(x)
= φc(x) , (12.6)
Furthermore, from the second equation in (12.6) we have
δ2W [J ]
δJ(y)δJ(x)
=
δφc(x)
δJ(y)
= iD(x, y) , (12.7)
where we used the fact that the second functional derivative of W [J ] must give us the
connected two-point function D(x, y). Now from the first equation in (12.6) we see that
δ
δJ(y)
δΓ[φc]
δφc(x)
= −δ(x− y) , (12.8)
from which we can write
δ(x− y) = −
∫
d4z
δφc(z)
δJ(y)
δ2Γ[φc]
δφc(z)δφc(x)
(12.9)
= −
∫
d4z
δ2W [J ]
δJ(y) δJ(x)
δ2Γ[φc]
δφc(z)δφc(x)
.
The expression above means that the first factor is the functional inverse of the second,
that is
− δ
2W [J ]
δJ(y) δJ(x)
=
(
δ2Γ[φc]
δφc(y) δφc(x)
)−1
. (12.10)
This tells us that the two-point connected correlation function is basically the functional
inverse of the second derivative of the effective action. We now want to relate a higher
order connected correlation function to the functional derivatives of the effective action.
For this purpose, we first notice that
3
δ
δJ(z)
=
∫
d4w
δφc(w)
δJ(z)
δ
δφc(w)
(12.11)
= i
∫
d4wD(z, w)
δ
δφc(w)
.
Then, we can write
δ3W [J ]
δJ(x) δJ(y) δJ(z)
= i
∫
d4wD(z, w)
δ
δφc(w)
δ2W [J ]
δJ(x) δJ(y)
(12.12)
= −i
∫
d4wD(z, w)
δ
δφc(w)
(
δ2Γ[φc]
δφc(x) δφc(y)
)−1
.
But the derivative of the functional inverse in the second line in (12.12) can be expressed
in terms of the derivative of the functional by using that for a given M with inverse such
that
MM−1 = 1, (12.13)
then we have that
δM−1
δa
= −M−1 δM
δa
M−1 . (12.14)
Then we arrive at
δ3W [J ]
δJ(x) δJ(y) δJ(z)
= −i
∫
d4w d4u d4v D(z, w)D(x, u)D(y, v)
δ3Γ[φc]
δφc(u) δφc(v) δφc(w)
.
(12.15)
The expression above gives us a relationship between the connected three-point correlation
function, on the left hand side and the third funcional derivative of the effective action.
Just by inspection of the right hand side we see that the triple functional derivative must
be the amputated three-point correlation function since it corresponds to the connected
one on the left hand side but stripped of the two-point functions of the three external
4 LECTURE 12. THE EFFECTIVE ACTION AS A GENERATING FUNCTIONAL
legs, here represented by D(z, w), D(x, u) and D(y, v). Then, noticing that the connected
three-piont function is given by
(−i)3 δ
3iW [J ]
δJ(x) δJ(y) δJ(z)
= − δ
3W [J ]
δJ(x) δJ(y) δJ(z)
, (12.16)
we have that
δ3iΓ[φc]
δφc(x) δφc(y) δφc(z)
= G
(3)
1PI(x, y, z) , (12.17)
where on the right hand side is the amputated or one-particle irreducible (1PI) three-point
function. This can be generalized to higher order correlation functions so that
δniΓ[φc]
δφc(x1) . . . δφc(xn)
= G
(n)
1PI(x1, . . . , xn) ≡ Γ
(n)(x1, . . . , xn) , (12.18)
where we renamed the 1PI n-point correlation functions for simplicity. The expression
above means that the effective action is the generating functional of the 1PI diagrams.
Then, we can expand it as
iΓ[φc] =
∑
n
1
n!
∫
d4x1 . . . d
4xn Γ
(n)(x1, . . . , xn)φc(x1) . . . φc(xn) . (12.19)
which is equivalent to (12.18). In order to compute the effective potential and make
contact with our previous results for it, we start by writing the 1PI correlation functions
in momentum space. We define them by the Fourier transform
Γ(n)(x1, . . . , xn) =
∫
d4k1
(2π)4
· · · d
4kn
(2π)4
ei(k1·x1+···+kn·xn) Γ(n)(k1, . . . , kn) (2π)
4 δ(4)(k1+· · ·+kn) .
(12.20)
Then we can write the effective action expansion in (12.19) as
iΓ[φc] =
∑
n
1
n!
∫
d4x1 . . . d
4xn
∫
d4k1
(2π)4
· · · d
4kn
(2π)4
(2π)2δ(4)(k1 + · · ·+ kn)
×e(k1·x1+···+kn·xn) Γ(n)(k1, . . . , kn)φc(x1) . . . φc(xn) , (12.21)
On the other hand, we may write the effective action as a derivative expansion. This is
5
Γ[φc] =
∫
d4x
{
−Veff.(φc) +
1
2
Z(φc)∂µφc∂
µφc + . . .
}
, (12.22)
where the effective potential is the zero-derivative term, the kinetic term (including field
renormalization) is the two-derivative term and the dots denote terms with a higher num-
ber of derivatives. The derivative expansion in (12.22) can be matched with a momentum
expansion in (12.21). Expanding about zero momenta (12.21) can be written as
iΓ[φc] =
∑
n
1
n!
∫
d4x1 . . . d
4xn
∫
d4k1
(2π)4
· · · d
4kn
(2π)4
∫
d4x ei(k1+···+kn)·x
×e(k1·x1+···+kn·xn)
{
Γ(n)(0, . . . , 0)φc(x1) . . . φc(xn) + . . .
}
, (12.23)
where we have replaced the momentum conservation delta function by its integral repre-
sentation and we have only written explicitly the first term in the momentum expansion
of Γ(n)(k1, . . . , kn) about zero momenta, i.e. Γ
(n)(0, . . . , 0). Noticing that
∫
d4ki
(2π)4
eiki·(xi+x) = δ(4)(xi + x) , (12.24)
we obtain
iΓ[φc] =
∑
n
1
n!
∫
d4x
{
Γ(n)(0, . . . , 0)φnc (x) + . . .
}
. (12.25)
Comparing with the first term in (12.22) we obtain that the effective potential is given by
Veff.(φc) =
∑
n
1
n!
iΓ(n)(0, . . . , 0)φnc (x) . (12.26)
This expression states that the effective potential is an expansion in the vacuum expec-
tation value of the field, φc(x), where the coefficients are fixed by the 1PI correlation
functions at zero external momentum. For instance, for a theory of a real scalar, at tree
level we have
iΓ(2)(0, 0) = µ2 , iΓ(4)(0, . . . , 0) = λ . (12.27)
Since at leading order only these two amplitudes are non-zero, the effective potential is
very simple and coincides with the classical potential. But if we want to go beyond leading
6 LECTURE 12. THE EFFECTIVE ACTION AS A GENERATING FUNCTIONAL
+ + + +
...
Figure 12.1: One loop contributions to the effective potential Veff..
order in computing Veff. we need to consider loop corrections to the Γ
(n)(0, . . . , 0). Thus,
we need to consider loop contributions to the zero-momentum amplitudes. And, as we
will see below, all powers of φc(x) will contribute starting at one loop.
12.1 The One Loop Effective Potential
We consider the usual real scalar theory
L = 1
2
∂µφ∂
µφ− 1
2
µ2φ2 − λ
4!
φ4 . (12.28)
To go beyond leading order in the calculation of Veff. we need to compute the one loop
contributions to the Γ(n)(0, . . . , 0). These are shown in Figure 12.1, where the φ4 interac-
tion is inserted. Since the theory respects φ → −φ, only diagrams with an even number
of external legs contribute. The infinite sum corresponds to adding all the one-loop con-
tributions to the zero external momentum amplitudes, i.e.
Γ(2)(0, 0) + Γ(4)(0, . . . , 0) + Γ(6)(0, . . . , 0) + Γ(8)(0, . . . , 0) + . . . . (12.29)
The expression for the one loop contribution to the zero-momentum 2n-point amplitude
is
Γ(2n)(0, . . . , 0) =
(2n)!
2n 2n
∫
d4k
(2π)4
[
(−iλ) i
k2 − µ2
]n
. (12.30)
In the expression above, there are n insertions of (−iλ) since there are n vertices. There
are also n propagators. Finally, the combinatoric factor outsidethe integral deserves a
more detailed explanation. First, the factor of (2n)! in the numerator corresponds to the
number of ways to distribute 2n particles in 2n external lines. However, some of these are
redundant. The factor of 2n in the denominator corresponds to the fact that exchanging
the two external lines at each of the n vertices does not result in a new contribution.
12.1. THE ONE LOOP EFFECTIVE POTENTIAL 7
Finally, the factor of 2n also in the denominator accounts for the indistinguishable overall
rotations of the vertices, which do not generate distinct diagrams. We can now use
(12.30) in the expression (12.26) for the effective potential, noticing that only terms even
contribute in it (i.e. the factorial dividing in (12.26) is (2n)! now). We then obtain
Veff.(φc) = V (φc) +
i
2
∫
d4k
(2π)4
∞∑
n=1
1
n
[
(λ/2)φ2c
k2 − µ2
]n
. (12.31)
It is straightforward to show that
∞∑
n=1
1
n
[
(λ/2)φ2c
k2 − µ2
]n
= − ln
(
1− (λ/2)φ
2
c
k2 − µ2
)
, (12.32)
which results in
Veff.(φc) = V (φc)−
i
2
∫
d4k
(2π)4
ln
(
1− (λ/2)φ
2
c
k2 − µ2
)
. (12.33)
The result above for the one-loop effective potential is very similar to the one we obtained
before, with the only difference that now we have a nonzero mass µ2. The method we
used here makes it clear that the corrections to the potential are coming first from the
sum of all the one loop contributions to the zero external momentum amplitudes.
As usual, we need to regulate the momentum integral and fix the relevant counterterms
by imposing renormalization conditions on them. After a Wick rotation with k0 = ik4 we
have
Veff.(φc) = V (φc) +
1
16π2
∫ Λ
0
dk k3 ln
(
1 +
(λ/2)φ2c
k2 + µ2
)
+
1
2
δµ2φ2c +
δλ
4!
φ4c , (12.34)
where we are already using the euclidean momentum and we added the only two necessary
counterterms since the integral has quadratic and logarithmic divergences coming from
the first two terms in the expansion of the logarithm. We impose the renormalization
conditions
d2Veff.(φc)
dφ2c
∣∣∣∣
φc=0
= µ2 , (12.35)
that fixes the physical mass, and
8 LECTURE 12. THE EFFECTIVE ACTION AS A GENERATING FUNCTIONAL
d4Veff.(φc)
dφ4c
∣∣∣∣
φc=0
= λ , (12.36)
for fixing the physical coupling. Performing the euclidean momentum integration with a
cutoff Λ, we have
Veff.(φc) = V (φc) +
1
64π2
{
λ
2
φ2cΛ
2 +
(
λ
2
φ2c + µ
2
)2
ln
(
(λ/2)φ2c + µ
2
Λ2
)
+ µ2 ln
(
µ2
Λ2
)
+
1
2
δµ2φ2c +
δλ
4!
φ4c
}
. (12.37)
Imposing (12.35) and (12.36) in (12.37) fixes the counterterms to be
δµ2 = − 1
64π2
{
λΛ2 + λµ2 + 2λµ2 ln
(
µ2
Λ2
)}
, (12.38)
and
δλ = − 1
64π2
{
9λ2 + 6λ2 ln
(
µ2
Λ2
)}
. (12.39)
Replacing these back in (12.37) we obtain the renormalized effective potential to one loop
accuracy,
Veff.(φc) = V (φc) +
1
64π2
{(
λ
2
φ2c + µ
2
)2
ln
(
(λ/2)φ2c + µ
2
µ2
)
− λµ
2
2
φ2c −
3
8
λ2φ4c
}
.
(12.40)
This expression is the same we obtained before but for nonzero µ2. Once again then we
conclude that the loop corrections to the potential not only shift the parameters, in this
case µ2 and λ, but also change the functional form of Veff.(φc).
The result above is independent of the sign of µ2. Thus, if µ2 < 0 we have spontaneous
symmetry breaking and the vacuum expectation value (VEV) of φ is nonzero. Then µ2 is
not the physical mass of the field φ. However, it is customary to define the VEV-dependent
mass as
m2(φc) ≡ µ2 +
λ
2
φ2c , (12.41)
12.1. THE ONE LOOP EFFECTIVE POTENTIAL 9
which then enters in the expression (12.40) for Veff.. This results in
Veff.(φc) = V (φc) +
1
64π2
m4(φc) ln
(
m2(φc)
µ2
)
+ · · · , (12.42)
where the dots denote the one-loop shifts of the φc powers already existing at leading
order. This form will be very handy when computing the effective potential of theories
with spontaneous symmetry breaking.
Additional suggested readings
• An Introduction to Quantum Field Theory, M. Peskin and D. Schroeder, Chapter
11.
• Quantum Field Theory in a Nutshell, by A. Zee. Chapter 4.3.
• Gauge Theory of Elementary Particle Physics, T.-P. Cheng and L.-F. Li, Section
6.4.